Success! Here's what we did:
First, it required two people - one (me) to hold the chicken and make encouraging remarks and soothing sounds, and the other to work with surgical precision (that would be my husband).
We soaked and soaked to no avail - the lumps weren't coming off. So my husband started to chip away at them with channel-lock pliers, only at the edges, never putting any pressure on the toenail. Once we got each lump down to a reasonable size (like a small marble), we soaked them some more. Then my husband cut away the lumps with a pair of side-cutter pliers with incredible care. Side-cutter pliers have a sharp point which helped to take off the side of a lump but required great precision. We decided that cutting away the lump was better than trying to pry it off, which would be hard on the toenail.
More soaking and the rest of each lump came off with our fingernails. The surgery on each chicken took about 15 minutes. They now have gorgeous clean nails and they are so very happily scratching in the grass and dirt.
I really like having chickens!
Dasha